Last updated on May 26th, 2025
If a number is multiplied by the same number, the result is a square. The inverse of the square is a square root. The concept of square roots extends into complex numbers when dealing with negative values. Here, we will discuss the square root of -320.
The square root is the inverse of the square of a number. Since -320 is negative, its square root is not a real number. Instead, we express it in terms of imaginary numbers. The square root of -320 can be expressed as √(-320) = √(320) × i, where i is the imaginary unit. In exponential form, it is expressed as (-320)^(1/2). √320 is approximately 17.8886, so √(-320) = 17.8886i, which is a complex number.
The square roots of negative numbers involve imaginary units. We use methods like prime factorization for positive components and then multiply by the imaginary unit. Let us now learn these methods:
We find the prime factors of 320 first because -320 has the same factors multiplied by the imaginary unit.
Step 1: Finding the prime factors of 320 Breaking it down, we get 2 x 2 x 2 x 2 x 2 x 5: 2^5 x 5
Step 2: Now that we have the prime factors of 320, we pair them to simplify. The simplification gives us √320 = √(2^4 × 2 × 5) = 4√20 = 8√5.
Step 3: Since -320 is negative, multiply the result by i. Therefore, the square root of -320 is 8√5i.
The long division method helps approximate the square root of positive numbers. For -320, we calculate the square root of 320 and then apply the imaginary unit.
Step 1: Group the digits of 320 from right to left. Here, it’s 32 and 0.
Step 2: Find the largest number whose square is less than or equal to 32. This number is 5 because 5 × 5 = 25.
Step 3: Subtract 25 from 32, giving a remainder of 7. Bring down the next digit, making the new dividend 70.
Step 4: Double the divisor, which is 10, and find a digit n such that (10n) × n is less than or equal to 70.
Step 5: Approximate further decimal places if needed. For √320, it approximates to 17.8886.
Step 6: Multiply by i for the negative square root: therefore, √(-320) = 17.8886i.
Approximation helps find the square root of a number swiftly. For -320, we calculate the square root of 320 and multiply by i.
Step 1: Determine the closest perfect squares around 320, which are 289 (17^2) and 324 (18^2). So, √320 is between 17 and 18.
Step 2: Use interpolation: (320 - 289) / (324 - 289) = 31 / 35 ≈ 0.886.
Step 3: Add this to the lower bound: 17 + 0.886 = 17.886.
Step 4: Multiply by i for the negative root: √(-320) = 17.886i.
Students often make mistakes when dealing with imaginary numbers, such as forgetting to include the imaginary unit i or incorrectly applying real number techniques. Let us examine some of these mistakes.
Can you help Max find the magnitude of a vector if its component is given as √(-45)?
The magnitude of the vector is 6.708i.
The magnitude involves the absolute value of the square root.
Here, √(-45) = √45 × i ≈ 6.708i.
Therefore, the magnitude is 6.708i.
A square-shaped field has an area of -320 square units. How would you describe the side length?
The side length is 17.8886i units.
For a negative area, we consider the imaginary side. √(-320) = 17.8886i.
Thus, the side length is 17.8886i units.
Calculate √(-320) × 5.
The result is 89.443i.
First, find √(-320) = 17.8886i.
Then, multiply by 5: 17.8886 × 5 = 89.443i.
What is the square root of (-100 + 20)?
The square root is 8.9443i.
First, calculate (-100 + 20) = -80.
Then, √(-80) = √80 × i = 8.9443i.
Find the perimeter of a rectangle with length 'l' as √(-128) units and width 'w' as 20 units.
The perimeter is 40 + 22.6274i units.
Perimeter of the rectangle = 2 × (length + width).
Perimeter = 2 × (√(-128) + 20) = 2 × (11.3137i + 20) = 40 + 22.6274i units.
Jaskaran Singh Saluja is a math wizard with nearly three years of experience as a math teacher. His expertise is in algebra, so he can make algebra classes interesting by turning tricky equations into simple puzzles.
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